(2x-1)(4x^2+2x+1)-8(x+2)(x^2-2x+4)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (2x-1)(4x^2+2x+1)-8(x+2)(x^2-2x+4)

This expression involves the multiplication of two binomials and two trinomials. Let's break it down step by step to simplify it.

Recognizing Special Forms

The trinomials in the expression are special forms. Notice that:

• (4x^2 + 2x + 1) is a perfect square trinomial, which can be factored as (2x + 1)^2.
• (x^2 - 2x + 4) is also a perfect square trinomial, but with a negative middle term, making it factor as (x - 2)^2.

Applying the Formulas

Now we can rewrite the expression using these factored forms:

(2x-1)(4x^2+2x+1)-8(x+2)(x^2-2x+4) = (2x-1)(2x+1)^2 - 8(x+2)(x-2)^2

Expanding and Simplifying

We can simplify the expression by expanding the squares and multiplying the terms:

1. Expand the squares:

   • (2x+1)^2 = (2x+1)(2x+1) = 4x^2 + 4x + 1
   • (x-2)^2 = (x-2)(x-2) = x^2 - 4x + 4

2. Substitute the expanded terms back into the expression: (2x-1)(4x^2 + 4x + 1) - 8(x+2)(x^2 - 4x + 4)

3. Multiply using the distributive property:

   • (2x-1)(4x^2 + 4x + 1) = 8x^3 + 8x^2 + 2x - 4x^2 - 4x - 1
   • 8(x+2)(x^2 - 4x + 4) = 8(x^3 - 4x^2 + 4x + 2x^2 - 8x + 8) = 8x^3 - 24x^2 + 32x + 16x^2 - 64x + 64

4. Combine like terms: 8x^3 + 4x^2 - 2x - 1 - 8x^3 - 8x^2 + 32x - 64

5. Simplify: -4x^2 + 30x - 65

Final Result

Therefore, the simplified expression is -4x^2 + 30x - 65.